Construction C ⋆ from Self-Dual Codes
Maiara Francine Bollauf, Sueli Irene Rodrigues Costa, Ram Zamir
aa r X i v : . [ c s . I T ] M a y Lattice Construction C ⋆ from Self-Dual Codes Maiara F. Bollauf ∗ , Sueli I. R. Costa ∗ , and Ram Zamir ‡∗ Institute of Mathematics, Statistic and Scientific ComputingUniversity of Campinas, S˜ao Paulo13083-859, BrazilEmail: [email protected], [email protected] † Deptartment Electrical Engineering-SystemsTel Aviv University, Tel Aviv, IsraelEmail: [email protected]
Abstract —Construction C ⋆ was recently introduced as ageneralization of the multilevel Construction C (or Forney’scode-formula), such that the coded levels may be dependent.Both constructions do not produce a lattice in general, hencethe central idea of this paper is to present a − level latticeConstruction C ⋆ scheme that admits an efficient nearest-neighborhood decoding. In order to achieve this objective, wechoose coupled codes for levels and , and set the secondlevel code C as an independent linear binary self-dual code,which is known to have a rich mathematical structure amongfamilies of linear codes. Our main result states a necessary andsufficient condition for this construction to generate a lattice. Wethen present examples of efficient lattices and also non-latticeconstellations with good packing properties. Index terms — Multilevel construction, Construction C, Con-struction C ⋆ , self-dual codes, sphere packing.I. I NTRODUCTION
A lattice is a well studied mathematical structure due to anextensive list of applications, including its efficient packingproperties. The sphere packing problem has known solutionsonly for dimensions , , and , [13], [6], [21] and all ofthem can be reached by lattices. For other dimensions, thereare strong beliefs that the best possible packing density canbe achieved by lattices.One way of producing lattice constellations is to use lin-ear codes in the so called Constructions A, B, and D [8].There are also other interesting constructions that generatemore general constellations (lattices and non-lattices) withprominent applications in quantization and coded modulation,such as Constructions C [11] and C ⋆ [4]. The advantage ofworking with such constructions is mainly the translation ofcharacteristics from the linear code over a finite field to aninfinite constellation in the n − dimensional real space.While the condition for Construction C to be a lattice iselegant and directly related to Construction D [15], the latticecondition for its generalization, i.e. Construction C ⋆ , cannotbe related to any other previous lattice construction [4]. Thus,one proposal of this work is to investigate families of codeswhich make Construction C ⋆ always a lattice and the resultpoints out to the role of self-dual codes. In coding theory, self-dual codes are of a peculiar impor-tance as they represent the best known error correcting codesfor transmission or data storage [14], when one is interestedin transmitting a large number of messages with a largeminimum weight, in order to correct maximum number oferrors. Their properties and relations with results from grouptheory, combinatorics and lattices are well known. Self-dualcodes underlying Construction A are explored in several works[2], [17], [19] regarding the association of these codes tounimodular lattices.We are inspired by the − level Construction C ⋆ of theLeech lattice presented in [4], which considered coupledcodes for levels and , while the second level was the [24 , , − Golay code. We generalize this idea for any evendimension by fixing the choice of the second level code C tobe a self-dual code and our main result states a necessary andsufficient condition for such construction to produce a lattice.This theory also arises as a promising approach for the openproblem of decoding Construction C ⋆ , by using an extensionof the works from Forney [12] and Amrani et al. [1] to any − level lattice Construction C ⋆ . We present alternative constructions for the E lattice andknown packings in dimension and . Interesting non-lattice constellations (with a code C which is not self-dual),including a special one in dimension 4 that achieves the samepacking density of the lattice D , are presented.This paper is organized as follows: Section II introducessome relevant notions about lattices, Construction C ⋆ , andcodes. Section III presents a general way of producing latticesvia a − level Construction C ⋆ by using self-dual codes inthe second level. Section IV is devoted to examples of latticepackings. Section V describes non-lattice constellations whichhave good packing properties, including one that presents thesame packing density as the densest known lattice in R . Finally, in Section VI conclusions and perspectives of futurework are drawn.II. B
ACKGROUND ON LATTICES AND CODES
In this section, we recall the definition of Construction C ⋆ and the condition for it to be a lattice. We also point out someproperties of self-orthogonal and self-dual codes. efinition 1. (Lattice) A lattice Λ ⊂ R N is a set of integerlinear combinations of independent vectors v , v , . . . , v n ∈ R N . We say that a lattice is full rank if N = n, which is the caseof lattices explored through this paper. The volume vol (Λ) ofa full rank lattice is the absolute value of the determinantof a matrix which has its columns as the generator vectors v , v , . . . , v n . Definition 2. (Packing radius and packing density) The pack-ing radius r pack (Λ) of a lattice Λ ⊂ R n is half of theminimum distance between lattice points and the packingdensity ∆(Λ) is the fraction of the space that is covered byballs B ( λ, r pack (Λ)) of radius r pack (Λ) , centered at a latticepoint λ ∈ Λ , i.e., ∆(Λ) = vol ( B (0 , r pack (Λ)) vol (Λ) = V n r npack vol (Λ) , (1) where V n refers to the volume of the unit ball in R n . The packing density is an important measure to comparelattices. However, for increasing dimensions, this value tendsto zero and analogies are hard to perform. In that case, insteadof analyzing packing densities it is common to compareHermite constants.
Definition 3. (Hermite constant) The Hermite constant of alattice Λ ⊂ R n is given by γ n (Λ) = 4 (cid:18) ∆(Λ) V n (cid:19) /n = 4 r pack vol (Λ) /n = d (Λ) vol (Λ) /n , (2) where V n refers to the volume of the unit ball in R n . The Hermite constant γ n measures the highest attainablecoding gain of an n − dimensional lattice.Besides the well known Constructions A and D, that pro-duce lattice constellations from linear codes, another interest-ing construction is the so called Construction C or constructionby code-formula [11]. Definition 4. (Construction C) Consider L binary codes C , . . . , C L ⊆ F n , not necessarily nested or linear. Thenwe define an infinite constellation Γ C in R n that is calledConstruction C as: Γ C := C + 2 C + · · · + 2 L − C L + 2 L Z n . (3)A generalization of Construction C was introduced in [3],[4] and denoted by Construction C ⋆ . It was inspired bybit-interleaved coded modulation (BICM) and asymptotically,it was demonstrated its superior packing efficiency whencompared to Construction C.The main feature of Construction C ⋆ that differs fromConstruction C is the fact that the levels are inter-coded, i.e.,they are dependent. Definition 5. (Construction C ⋆ ) Let C ⊆ F nL be a binarycode. Then Construction C ⋆ is defined as Γ C ⋆ := { c + 2 c + · · · + 2 L − c L + 2 L z : ( c , c , . . . , c L ) ∈ C ,c i ∈ F n , i = 1 , . . . , L, z ∈ Z n } . (4) Note that Construction C coincides with Construction C ⋆ when C = C ×· · ·×C n and we observe that both constructionsin general do not produce a lattice. A condition that will assurethe laticeness of Construction C ⋆ will be presented next. Definition 6. (Projection codes) Let c = ( c , ..., c L ) be apartition of a codeword c = ( c , . . . , c n , ...., c L , . . . , c Ln ) ∈C ⊆ F nL into length − n subvectors c i = ( c i , ...., c in ) , i =1 , . . . , L. Then, a projection code C i consists of all subvectors c i that appear as we scan through all possible codewords c ∈ C . In what follows, we denote by + the real addition and by ⊕ the sum in F , i.e., x ⊕ y = ( x + y ) mod 2 . Definition 7. (Antiprojection) The antiprojection S i ( c , . . . ,c i − , c i +1 , . . . , c L ) consists of all vectors c i ∈ C i , i = 1 , . . . , L that appear as we scan through all possible codewords c ∈ C , while keeping c , . . . , c i − , c i +1 , . . . , c L fixed: S i ( c , ..., c i − , c i +1 , ..., c L ) = { c i ∈ C i :( c , . . . , c i |{z} i-th position , . . . , c L ) ∈ C} . (5)In [4], there are two statements that guarantee the latticenessof Construction C ⋆ and here we recall one of them, due to itssimplicity and straightforward relation with the results of thispaper. We start by the definition of Schur product. Definition 8. (Schur product) For x = ( x , . . . , x n ) and y =( y , . . . , y n ) both in F n , we define x ∗ y = ( x y , . . . , x n y n ) . Consider ψ : F n → R n as the natural embedding. Then, for x, y ∈ F n , it is valid that ψ ( x ) + ψ ( y ) = ψ ( x ⊕ y ) + 2 ψ ( x ∗ y ) . (6)In order to simplify, we abuse the notation, writing Eq. (6) as x + y = x ⊕ y + 2( x ∗ y ) . (7)A chain C ⊆ C ⊆ F n is said to be closed under Schurproduct if for any c , ˜ c ∈ C , the Schur product c ∗ ˜ c ∈ C . Theorem 1. [4] (A sufficient lattice condition for Γ C ⋆ ) If C ⊆ F nL is a linear binary code with projection codes C , C , . . . , C L such that C ⊆ S (0 , . . . , ⊆ C ⊆ · · · ⊆C L − ⊆ S L (0 , . . . , ⊆ C L ⊆ F n and the chain C i − ⊆S i (0 , . . . , is closed under the Schur product for all i =2 , . . . , L, then Γ C ⋆ is a lattice. In this paper we set L = 3 for Construction C ⋆ and analyzethe case where the second level code C is a self-orthogonallinear code in F n , independent of the other two levels. In F , the standard inner product of c = ( c , c , . . . , c n ) and ˜ c = (˜ c , ˜ c , . . . , ˜ c n ) is defined as h c, ˜ c i = P ni =1 c i ˜ c i mod 2 and the orthogonal set C ⊥ of a code C ⊆ F n is also definedas the set C ⊥ = { c ∈ F n : h c, ˜ c i = 0 , ∀ ˜ c ∈ C} . Definition 9. (Self-orthogonal and self-dual codes) A code C is self-orthogonal if C ⊂ C ⊥ and it is self-dual if C = C ⊥ . code C is self-orthogonal if and only if h c, ˜ c i = 0 , forall c, ˜ c ∈ C . Each codeword in a self-orthogonal code haseven Hamming weight and (1 , . . . , ∈ C ⊥ . Indeed, let c ∈C , which is a self-orthogonal code, then h c, c i = 0 , and itmeans that the Hamming weight of c, i.e. ω ( c ) , is alwayseven for all c ∈ C . Also, (1 , . . . , ∈ C ⊥ due to the fact that h c, (1 , . . . , i = 0 , for all c ∈ C and ω ( c ) is even.A characterization of self-dual codes is given by [10, p.8][16]: a [ n, k, d ] − linear code C is self-dual if and only if C ⊂ C ⊥ and k = n . Example 1.
The Reed-Muller code RM (1 , , whichis a [16 , , − binary linear code is self-orthogonal,while the [8 , , − extended Hamming code and the [24 , , − extended Golay code are both examples of self-dual codes. III. G
ENERAL LATTICES VIA LEVEL C ONSTRUCTION C ⋆ Inspired by the Leech lattice construction via C ⋆ presentedin [3], we aim to describe a more general − level latticeConstruction C ⋆ by fixing the level (projection) codes as • C = { (0 , . . . , , (1 , . . . , } ⊂ F n , which is the repeti-tion code; • C ⊂ F n as a convenient code we are going to explorelater; • C = ˜ C ∪C = F n , and we require that if c = (0 , . . . , then c ∈ ˜ C = { ( x , . . . , x n ) ∈ F n : P ni =1 x i ≡ } and if c = (1 , . . . , then c ∈ C = { ( y , . . . , y n ) ∈ F n : P ni =1 y i ≡ } . In other words, the main code
C ⊆ F n is given by C = { (0 , . . . , | {z } ∈C , a , . . . , a n | {z } ∈C , x , . . . , x n | {z } ∈ ˜ C ) , (1 , . . . , | {z } ∈C , a , . . . , a n | {z } ∈C , y , . . . , y n | {z } ∈C ) } . (8)One can notice that the dependence between levels is crucialin the definition of the main code C ⊆ F n , as in Eq. (8). Wecan then define a constellation Γ C ⋆ as the − level Construction C ⋆ given by Γ C ⋆ = { c + 2 c + 4 c + 8 z : ( c , c , c ) ∈ C , z ∈ Z n } . (9)The choice of C in Eq. (8) is directly related to Theorem1, as we are interested in constructing lattice constellations. Theorem 2. (Lattice Construction C ⋆ with self-orthogonalcodes) Let C ⊂ F n be a linear code according to Eq. (8) . Theresulting constellation Γ C ⋆ (Eq. (9) ) obtained via Construction C ⋆ from the code C is a lattice if and only if C ⊆ F n is aself-orthogonal code that contains (1 , . . . , . Proof. ( ⇒ ) Suppose that Γ C ⋆ constructed from C ⊆ F n is alattice. Then, given x, y ∈ Γ C ⋆ it is true that x + y ∈ Γ C ⋆ . We can write x = c + 2 c + 4 c + 8 zy = c ′ + 2 c ′ + 4 c ′ + 8 z ′ and x + y ∈ Γ C ⋆ implies that the vector ( c ⊕ c ′ , c ⊕ c ′ ⊕ ( c ∗ c ′ ) ,c ⊕ c ′ ⊕ (( c ∗ c ′ ) ∗ ( c ⊕ c ′ ) ⊕ ( c ∗ c ′ )) ∈ C (10)and in particular, c ⊕ c ′ ⊕ ( c ∗ c ′ ) ∈ C . Due to linearity, c ⊕ c ′ ∈ C and for c = c ′ = (1 , . . . , , we must have that (1 , . . . , ∈ C . It remains to demonstrate the C is self-orthogonal. Thereare only four possible choices for c and c ′ , which we discusscase by case below: • c = c ′ = (0 , . . . ,
0) : from Eq. (10) we have that (0 , . . . , , c ⊕ c ′ , c ⊕ c ′ ⊕ c ∗ c ′ ) ∈ C , where by con-struction c ⊕ c ′ has even weight, so it is straightforwardto conclude that the sum of the coordinates of c ∗ c ′ isequal to zero and h c , c ′ i = 0 . • c = (1 , . . . , and c ′ = (0 , . . . ,
0) : from Eq. (10) wehave that (1 , . . . , , c ⊕ c ′ , c ⊕ c ′ ⊕ c ∗ c ′ ) ∈ C , whereby construction the coordinates of c sum one modulo 2and the coordinates of c ′ sum zero modulo 2, thus theonly possibility is that the sum of c ∗ c ′ is equal to zeroand h c , c ′ i = 0 . An analogous argument applies to thecase where c = (0 , . . . , and c ′ = (1 , . . . , . • c = c ′ = (1 , . . . ,
1) : from Eq. (10) we have that (0 , . . . , , c ⊕ c ′ ⊕ (1 , . . . , , c ⊕ c ′ ⊕ ( c ⊕ c ′ ) ⊕ c ∗ c ′ ) ∈C , where in this case both coordinates of c and c ′ sum one modulo 2, hence c ⊕ c ′ has even weightand consequently also ( c ⊕ c ′ ) ⊕ c ∗ c ′ must haveeven weight. We need to prove that the coordinates of c ∗ c ′ sum zero modulo 2. Assume that c ⊕ c ′ has oddweight, by contradiction (because it will force c ∗ c ′ tohave odd weight as well). Due to the linearity of C ,c ⊕ c ′ = ˜ c ∈ C . Then, we consider in Eq. (10), c = c ′ = ˜ c , which yields: (0 , . . . , , , . . . , , c ⊕ c ′ ⊕ ˜ c ⊕ ˜ c ⊕ ˜ c ∗ ˜ c ) ∈ C , (11)and (˜ c ⊕ ˜ c ) ⊕ (˜ c ∗ ˜ c ) = ˜ c , what makes the thirdcoordinate to have odd weight. Thus, the element writtenin Eq. (11) does not belong to the code C and we havea contradiction. Therefore, both c ⊕ c ′ and c ∗ c ′ musthave even weight, what implies that h c , c ′ i = 0 . We can then conclude that C is self-orthogonal. ( ⇐ ) To assure the latticeness condition from Theorem 1 tohold one needs to first verify that C ⊆ S (0 , . . . , ⊆ C ⊆ S (0 , . . . , ⊆ C , (12)and due to the structure of C ⊆ F n in Eq. (8) we havethat S (0 , . . . ,
0) = C and S (0 , . . . ,
0) = ˜ C . By hy-pothesis, (1 , . . . , ∈ C , what allow us to conclude that C ⊆ S (0 , . . . , and this nesting is clearly closed underSchur product.Since C is self-orthogonal, all codewords have even weightand C ⊆ ˜ C . It remains to show that this nesting is closedunder Schur product, i.e., given any c , c ′ ∈ C , the sum ofall coordinates of the vector defined by c ∗ c ′ should be zeroodulo 2. Observe that the Schur product is the coordinate-by-coordinate product and the action of summing all componentsof the resulting Schur product vector is the same as h c , c ′ i . Thus, we want to prove that h c , c ′ i = 0 mod 2 , which is truesince C is self-orthogonal.One can observe that for self-dual codes, the conditionrequired by Theorem 2 is automatically satisfied, because C = C ⊥ and also (1 , . . . , ∈ C ⊥ . IV. C
ONSTRUCTIONS OF KNOWN LATTICES VIA C ⋆ We can only expect to have interesting lattice constellationsvia Construction C ⋆ following the procedure described inSection III for n even, because we need to assure that (1 , . . . , ∈ C ⊆ S (0 , . . . ,
0) = ˜ C . This section summarizes some new lattice constructionsfor even dimensions built from a − level Construction C ⋆ with the main code C ⊆ F n as in Eq. (8), whose resultingconstellation is Γ C ⋆ as in Eq. (9).Observe that an essential feature to calculate the packingefficiency or Hermite constant of a lattice is the minimumdistance. A closed formula for the minimum distance of aconstellation generated by Construction C ⋆ is still an openproblem and in general, what is known is just an upper andlower bound for it [4]. However, for particular cases, when thecodes are established, as it is the case of the examples exploredin this section, this calculation can be done by brute force, i.e.,by investigating all possible minimum weight codewords andcalculating the minimum among them. Dimension 8 - E lattice: Define C as the [8 , , − extendedHamming code, which is self-dual and whose basis vectorsare displayed in the rows of the following generator matrix, G = . (13)One can notice that the minimum distance of C is , of C is , and of ˜ C and C is . Then, because of the dependencecreated by the main code C (Eq. (8)), in order to calculatethe squared minimum distance of Γ C ⋆ , we may consider thecombinations of codewords that yields in the minimum, i.e., d (Γ C ⋆ ) = min { d H ( C ) + 2 d H ( C ) , d H ( C ) , d H ( ˜ C ) , d H ( C ) } = min { , · , · , · } = 16 and d min (Γ C ⋆ ) = 4 . Here, d H denotes the minimum Hammingweight of the respective code. Hence, the packing density ofthis construction is calculated by ∆(Γ C ⋆ ) = |C| vol ( B (0 , d min ))2 n = 2 · · π
4! 2 ≈ . , (14)which coincides with the packing density of the E latticeand E = √ Γ C ⋆ . This construction is just to illustrate that one can achieve the same packing density as E latticevia Construction C ⋆ , although the most efficient way ofrepresenting this lattice is via Construction A. Dimension 14:
Consider C as the self-dual code [14 , , . Thus, d (Γ C ⋆ ) = min {
32 + 14 , · , · } = 16 , whose Hermite constant is γ (Γ C ⋆ ) = d (Γ C ⋆ ) vol (Γ C ⋆ ) /n = 16(2 ) / = 2 . (15)The upper bound for the Hermite constant in this dimensionis . , according to [5].In dimension , the best known packing density is givenby the decoupled version of Eq.(5), where C = RM (2 , , where RM ( r, m ) denotes the Reed-Muller code of length m and order r. In this particular case, Construction C, D and C ⋆ coincides. Dimension 24: (Leech lattice) This construction was al-ready presented in [3], [4] and it assumes C as the [24 , , − extended Golay code. Dimension 32:
Define C as the RM (2 , , which is a [32 , , − self-dual code. Then, we have that, following ananalogous calculation for the minimum distance as it was donein the E case, d (Γ C ⋆ ) = min {
32 + 16 , · , · } = 32 . Hence, the Hermite constant is γ (Γ C ⋆ ) = d (Γ C ⋆ ) vol (Γ C ⋆ ) /n = 32(2 ) / = 4 , (16)which coincides with Hermite constant of the Barnes-Walllattice BW . Dimension 40:
Define C as an extremal self-dual [40 , , − code, i.e., its minimum distance achieves the high-est possible value for given k and n. The squared minimumdistance is given by d (Γ C ⋆ ) = min {
40 + 16 , · , · } = 32 . The Hermite constant of this lattice constellation is γ (Γ C ⋆ ) = d (Γ C ⋆ ) vol (Γ C ⋆ ) /n = 32(2 ) / = 4 , (17)which coincides the Hermite constant given by the extremaleven unimodular lattice in dimension . V. S
PECIAL NON - LATTICE CONSTELLATIONS
One can notice that the scheme proposed for Construction C ⋆ may be also used to get non-lattice constellations whenthe code C is not self-orthogonal or it is but does not containthe codeword (1 , . . . , . Dimension 4:
It is believed that the best known packingdensity for any constellation in dimension n = 4 is given byhe lattice D [7], [8], which is, up to congruence, the uniquelattice that achieves this density. In the sequel, we presenta non-lattice constellation that achieves the same packingdensity as D . We consider C and C as the coupled codes according toSection III, and C is the RM (1 ,
2) [4 , , − code, i.e., RM (1 ,
2) = { (0 , , , , (1 , , , , (0 , , , , (1 , , , , (0 , , , , (0 , , , , (1 , , , , (1 , , , } , we can see that this code is not self-orthogonal. Moreover, ifwe apply a Construction C ⋆ as proposed in Eq. (9), it does notgive a lattice. Indeed, consider (4 , , , , (4 , , , ∈ Γ C ⋆ . Their real sum is (8 , , ,
4) = (0 , , ,
0) + 2(0 , , ,
0) +4(0 , , ,
1) + 8(1 , , , and (0 , , , , , , , , , , , / ∈C ⊆ F . When we calculate the squared minimum distanceof this constellation, we have that d (Γ C ⋆ ) = min { , · , · } = min { , , } = 8 and d min (Γ C ⋆ ) = 2 √ . The packing density of this construc-tion is then ∆(Γ C ⋆ ) = |C| vol ( B (0 , d min ))2 n = 2 · · π
2! ( √ = π ≈ . ... which is the same packing density as the D lattice.Other interesting non-lattice cases obtained by an analogousconstruction are the following: Dimension 18:
Considering C to be the [18 , , − binarylinear code [18], the resulting constellation achieves the bestknown Hermite constant in this dimension [5]. Dimension 20:
The best sphere packing in dimension ispresented in the work of Vardy [20] and it can be seen as aConstruction C ⋆ , where the three levels are coupled. Dimension 40:
By assuming C as the [40 , , − binarylinear code [8, p. 146], we can slightly improve the Hermiteconstant of the lattice presented in Section IV in dimension , which in this case reaches γ = 4 . . VI. C
ONCLUSION AND FUTURE WORK
We detailed some lattice constructions under the perspectiveof a special scheme of Construction C ⋆ , using coupled firstand third levels and admitting as second level self-dual codes.This construction is only interesting for low dimensions,because the choice of the most significant bit code (third level)forces an upper bound for the squared minimum distance equalto , which does not depend on the dimension. This drawbackmay be solved by applying Construction C ⋆ to other familiesof coupled codes or by increasing the number of levels.We also presented non-lattice constructions, including a fourdimensional Construction C ⋆ that achieves the same packingdensity as the D lattice and interesting potentially interesting results for dimensions , , and . We aim in a future workto apply other self-dual codes to Construction C ⋆ , also withdifferent alphabet sizes, and compare it with known results forConstruction A [17].In terms of efficient decoding, the idea is to generalizethe bounded-distance decoding scheme for the Leech latticeproposed by Forney [12] to any − level lattice Construction C ⋆ built according the structure proposed by this paper.A CKNOWLEDGMENT
The authors would like to thank Joseph J. Boutros forfruitful discussions and also the reviewers for meaningfulsuggestions. SIRC was supported by CNPq (313326/2017-7) and FAPESP (2013/25977-7) Foundations, and RZ wassupported by Israel Science Foundation (676/15).R
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